General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux

Details General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux

1997-09-02

arxiv.org/abs/dg-ga/9709007v1

Mathematics

Differential Geometry

27 Pages, LaTeX 2.09. See also dg-ga/9709006

Scientific paper

For each end of complete minimal surface in the Euclidean 3-space, the flux vector is defined. It is well-known that the sum of the flux vector over all ends are zero. Consider the following inverse problem: For each balanced n-vectors, find an n-end catenoid which realizes these vectors as flux. Here, an n-end catenoid is a complete minimal surface of genus zero with ends asymptotic to the catenoids. In this paper, we show that the inverse problem can be solved for almost all balanced n vectors for arbitrary n, which is grater than 4. The assumption "almost all" is needed because nonexistence is known for special balanced vectors. We remark that in the case of n=4, the same result has been obtained by the authors (dg-ga/9709006). And the case n=3 is treated by Lopez and Berbanel.

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